Genus of The Curve
The degree of the polynomial determines the genus of the curve: a polynomial of degree 2g + 1 or 2g + 2 gives a curve of genus g. When the degree is equal to 2g + 1, the curve is called an imaginary hyperelliptic curve. Meanwhile, the curve that has degree 2g + 2 is mentioned a real hyperelliptic curve. This statement about genus remains true for g = 0 or 1, but those curves are not called "hyperelliptic". Rather, the case g = 1 (if we choose a distinguished point) is an elliptic curve. Hence the terminology.
Read more about this topic: Hyperelliptic Curve
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